West Point cadets in this period went through a four-year curriculum, where those in their final year were referred to as first years. About seventy-percent of classroom hours were devoted to math, science (called natural and experimental philosophy,) and engineering. Fourth and third year students took an hour and a half of math six days a week with the most advanced portion of the class advancing through integral calculus. Most other subjects were scattered throughout the four years. The sciences got specific, for example, an hour-long weekday instruction block for second years could encompass astronomy, optics, acoustics, and magnetism. The remainder of the curriculum was made up of French, drawing, rhetoric, more and more English as the period progressed, geography, political science and of course, tactics.
Classes were based on recitation, either oral or written. Students would spend class time reading aloud, working on problems, or explaining their process on the blackboard. Even lecture was rare and lessons were mostly theoretical. Rifle practice was limited to brief rotations in the fourth, second, and first classes, when students shot rounds at a 6 by 3 ft. targets from 150 yards. Professor H.C. Bartlett, instructor of my example second year class, was a notable exception to this rule. He offered demonstrations and forced students to take measurements with a sextant and learn to use a zenith telescope to measure astronomical latitude, which they passionately hated. There was a slight movement towards more practical lessons: horsemanship was practiced, as well as artillery, infantry, cavalry and fencing drills.
The case of textbooks is less clear. Bartlett wrote his own optics and acoustics textbook in 1839 for lack of anything else to his liking. It was later published as Elemental of Analytical Mechanics and used around the country. Dennis Hart Mahan, engineering professor and superintendent, is credited with most of the high level military engineering and tactical instruction during the period. He used many of his own writings as well and was influenced by French precedent. There has been some debate over what foreign theorists the cadets may or may not have read (almost always in the context of their decisions in the Civil War) without a better conclusion than "maybe Jomini, almost certainly not Clausewitz."
For a more detailed look at the curriculum, professors, and students, there's the oddly-structured but thorough "The Best School”: West Point 1833-1866 by James L. Morrison. After that, I don't know of other works encompassing the topic and can only recommend looking to individual biographies.
(And yes, General Buford graduated from West Point in 1848.)
I'll follow /u/General_Buford's excellent comment with a recommendation of The Class of 1846 by John Waugh, which covers life at the Point during this period.
In the fall of 1843, for example, a junior cadet could expect to study analytical geometry and calculus, surveying fieldwork, French, English grammar, rhetoric, geography, history, drawing and the first parts of artillery.
Drawing might seem unusual to modern eyes, but it had a very practical use for 19th century engineers, who didn't have access to modern photographic techniques. Per Waugh:
Perverse as it seemed, drawing, like French, had a purpose; an engineer must know how to draw, just as he must know how to read French.
From 1834, drawing was taught by Robert Weir, one of a handful of faculty at West Point with an international reputation. By 1843, Weir was ensconced as one of the leading lights of the Hudson River school of art.
William Sherman reports that in 1835, a year before he would be eligible for appointment to the Academy, he devoted himself to mathematics and French, “which were known to be the chief requisites for admission to West Point.” He attended from 1836 to 1840, and he recalled his principal professors as coming from the fields of engineering (Mahan), natural philosophy (Bartlett), chemistry, mathematics, drawing, and French. In his memoirs he recalls:
“The routine of military training and of instruction was then fully established, and has remained almost the same ever since. To give a mere outline would swell this to an inconvenient size, and I therefore merely state that I went through the regular course of four years, graduating in June, 1840, number six in a class of forty-three. These forty-three were all that remained of more than one hundred which originally constituted the class. At the Academy I was not considered a good soldier, for at no time was I selected for any office, but remained a private throughout the whole four years. Then, as now, neatness in dress and form, with a strict conformity to the rules, were the qualifications required for office, and I suppose I was found not to excel in any of these. In studies I always held a respectable reputation with the professors, and generally ranked among the best, especially in drawing, chemistry, mathematics, and natural philosophy. My average demerits, per annum, were about one hundred and fifty, which reduced my final class standing from number four to six.”
Ulysses Grant attended several years later, beginning in 1839. He begins his account of his cadet career before regular classes began: “The encampment which preceded the commencement of academic studies was very wearisome and uninteresting. When the 28th of August came -- the date for breaking up camp and going into barracks -- I felt as though I had been at West Point always, and that if I staid to graduation, I would have to remain always.”
Grant continues: “I did not take hold of my studies with avidity, in fact I rarely ever read over a lesson the second time during my entire cadetship. I could not sit in my room doing nothing. There is a fine library connected with the Academy from which cadets can get books to read in their quarters. I devoted more time to these, than to books relating to the course of studies … Cooper’s Marryat’s Scott’s, Washington Irving’s works, Lever’s, and many others that I do not now remember. Mathematics was very easy to me, so that when January came, I passed the examination, taking a good standing in that branch. In French, the only other study at that time in the first year’s course, my standing was very low. In fact, if the class had been turned the other end foremost I should have been near head. I never succeeded in getting squarely at either end of my class, in any one study, during the four years. I came near it in French, artillery, infantry and cavalry tactics, and conduct.”
So there you have it: lots of worries about French and math, conduct was a problem for our two famous generals (as it was for Edgar Allen Poe, who dropped out in 1831), tactics were okay, and then there was a smattering of more forgettable subjects, such as natural philosophy, chemistry, and drawing. Incidentally, French was a necessity since military minds were debating Napoleon’s tactics, and West Point’s superintendent Col. Sylvanus Thayer had collected many of these texts to be the core of the academy library. Drawing (or drafting) was useful for engineers and artillerists (like Napoleon), who needed it to lay out defensive works or plan siege trenches.
Both Sherman and Grant, however, spend more time discussing their travels to and from West Point rather than their actual course of instruction. Apparently there was little love lost. Grant remembered from his first (a.k.a. “plebe”) year at the Academy: “Early in the session of the Congress which met in December, 1839, a bill was discussed abolishing the Military Academy. … I was selfish enough to favor the bill. It never passed, and a year later, although the time hung drearily with me, I would have been sorry to have seen it succeed.”
Both volumes are available in a single set published by the Library of America, which is the edition quoted above.